Differentiable stability and sphere theorems for manifolds and Einstein manifolds with positive scalar curvature

TL;DR

This paper develops differentiable stability and sphere theorems for manifolds with positive scalar curvature, extending Green's rigidity results and applying them to Einstein manifolds using convergence techniques.

Contribution

It introduces new differentiable stability and sphere theorems for positive scalar curvature manifolds, utilizing $C^{k,eta}$ convergence methods and extending Green's results to Einstein manifolds.

Findings

Proved differentiable stability theorems for manifolds with positive scalar curvature.

Extended sphere theorems to Einstein manifolds with positive scalar curvature.

Applied convergence techniques to establish rigidity results.

Abstract

Leon Green obtained remarkable rigidity results for manifolds of positive scalar curvature with large conjugate radius and/or injectivity radius. Using $C^{k,\alpha}$ convergence techniques, we prove several differentiable stability and sphere theorem versions of these results and apply those also to the study of Einstein manifolds.